
where \(v^\text{S}_l(\beta ) = \int _0^{2\pi } (d\phi /2\pi ) e^{il \phi } \, V_{\text{IAAF}}(\beta ,\tau ,x, \phi )\) . The model describes particles hopping on a 2D square lattice in the presence of a perpendicular magnetic field with \(2\pi /\tau \)  flux threading each square plaquette written in the Landau gauge using Peierls substitution [1]}, [2]}, see Fig. REF (a). Along the \(x\) -axis, we have nearest-neighbor hopping as in the descendent 1D model. Along the \(y\) -axis, we have long-ranged hopping depending on \(\beta \) , where in the limit of \(\beta \rightarrow 0\) , it also becomes short-ranged and we obtain the 2D model Hofstader model [3]}, [4]}, [5]}
\( H_{\rm Hof} = \sum _{n,m} \Big [ t c_{n,m}^\dag c_{n + 1,m}+ \frac{\lambda ^{\text{d}}}{2} e^{i2\pi n/\tau } c_{n,m}^\dag c_{n,m + 1} + h.c. \Big ]\,. \) 
<FIGURE>